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Complexity theory and combinatorial optimization Class #2 – 17 th of March

Complexity theory and combinatorial optimization Class #2 – 17 th of March. …. where we deal with decision problems, finite automata, Turing machines pink dogs, …. But also P, NP, NP-completeness, …. Introduction to computational intractability.

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Complexity theory and combinatorial optimization Class #2 – 17 th of March

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  1. Complexity theory and combinatorial optimization Class #2 – 17th of March …. where we deal with decision problems, finite automata, Turing machines pink dogs, …. But also P, NP, NP-completeness, …..

  2. Introduction to computational intractability Is my problem efficiently solved by a computer? an automatic machine an algorithm?

  3. What’s a problem? That is the problem …. Decision problems: each instance is a question Formal definition with language theory encoding scheme More natural problems: “meta-language”

  4. What’s an algorithm? Will be ever exist any pink dog outside London? • The pink dog question Does a pink dog exist? Does a pink dog exist (outside London)? Since the answer is yes, it can be answered Since there are only a finite number of dogs (outside London) and since for each one a can decide whether it is pink or not it can be answered. To answer it one needs a formal model of dogs.

  5. What’s an algorithm? • The pink dog question • Computability models • Lambda-calculus (A. Church, 1931) • General recursive functions (K. Gödel, 1934) • Turing machines (A. Turing, 1936) • Random-Access Machines, … • Church thesis Before the first computer

  6. What’s an algorithm: the Turing machine model • From finite states automaton ….. to Turing machines • 1-tape (deterministic) Turing Machine (DTM) • multi-tape Turing machines • non-deterministic Turing machines (NDTM) • transition function transition relation • put non-determinism at the beginning • equivalence between all these Turing machines models • universal Turing Machine Example

  7. Problems solved by Turing machines M: DTM, L a language on the same input alphabet Decision problem solved by an algorithm? (through an encoding scheme) • Is L recognized by M? • Is L decided by L? • The halting problem: an example of undecidable problem

  8. Complexity of Turing machines A notion of efficiency • Complexity of DTM (halting for each instance) • Complexity of NDTM • Polynomial-time: considered as efficiency (Cobham-Edmond’s thesis) • Difference between DTM and NDTM (from complexity point of view) • From languages to problems (reasonable encoding schemes)

  9. P, NP and NP-completeness • The class P • The class NP • Exponentially solving problems in NP • Polynomial reductions • NP-complete problems

  10. Some NP-complete problems • SAT • Cook’s theorem (1971) • How to prove NP-completeness after Cook? • 3-SAT • to be continued during the next class

  11. Enjoy your vacation

  12. An Example of 1-tape Deterministic Turing Machine q2 q1 q1 q0 qN b b 0 0 1 0 1 b b b b …

  13. q2 q2 q1 q1 q0 q2 b 0 0 1 1 1 b b b b … b

  14. q3 q4 q2 q4 q4 q4 b b 0 1 1 1 b b b b b …

  15. q2 q1 q2 q0 q4 b 0 1 1 b b b b b … b b

  16. q4 q4 q3 q2 b 1 1 b b b b b … b b b

  17. q4 q0 qN b 1 b b b b b b … b b

  18. q0 b 0 0 1 1 b b b b b …

  19. q3 q4 q2 q4 q4 b b 0 1 b 1 b b b b b …

  20. q4 q0 q1 q2 b 0 1 b b b b b b … b b

  21. q3 q3 q2 b b 1 b b b b b b … b b

  22. q3 q0 qY b b b b b b b b b b … What is the accepted language?

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